Questions: Write a polynomial f(x) that satisfies the given conditions. Express the polynomial with the lowest possible leading positive integer coefficient. Polynomial of lowest degree with lowest possible integer coefficients, and with zeros 8-6i and 0 (multiplicity 3). f(x)=

Write a polynomial f(x) that satisfies the given conditions. Express the polynomial with the lowest possible leading positive integer coefficient. Polynomial of lowest degree with lowest possible integer coefficients, and with zeros 8-6i and 0 (multiplicity 3). f(x)=
Transcript text: Write a polynomial $f(x)$ that satisfies the given conditions. Express the polynomial with the lowest possible leading positive integer coefficient. Polynomial of lowest degree with lowest possible integer coefficients, and with zeros $8-6 i$ and 0 (multiplicity 3). $f(x)=$
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Solution

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Solution Steps

Step 1: Constructing the Polynomial

Given the zeros \( 8 - 6i \) and \( 0 \) (with multiplicity 3), we can express the polynomial \( f(x) \) as follows:

\[ f(x) = (x - (8 - 6i))(x - (8 + 6i))(x - 0)^3 \]

Step 2: Expanding the Polynomial

Upon expanding the polynomial, we obtain:

\[ f(x) = x^5 - 16x^4 + 100x^3 \]

Final Answer

The polynomial \( f(x) \) that satisfies the given conditions is:

\[ \boxed{f(x) = x^5 - 16x^4 + 100x^3} \]

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